PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 94 (108) (2013), 205–217
DOI: 10.2298/PIM1308205M
ON RICCI TYPE IDENTITIES IN MANIFOLDS
WITH NON-SYMMETRIC AFFINE CONNECTION
Svetislav M. Minčić
Abstract. In [18], using polylinear mappings, we obtained several curvature
tensors in the space LN with non-symmetric affine connection ∇. By the same
method, we here examine Ricci type identities.
1. Introduction
Consider N -dimensional differentiable manifold MN on which a non-symmetric
1
affine connection ∇ is defined. If X(MN ) is a Lie algebra of smooth vector fields
2
and X, Y ∈ X(MN ), then the mapping ∇: X(MN ) × X(MN ) → X(MN ) given by
2
(1.1)
1
∇X Y = ∇Y X + [X, Y ]
2
defines an other non-symmetric connection ∇ on MN [14]. That means that we
have
θ
θ
θ
θ
θ
θ
∇Y1 +Y2 X = ∇Y1 X + ∇Y2 X,
θ
θ
∇f Y X = f ∇Y X,
θ
∇Y (X1 + X2 ) = ∇Y X1 + ∇Y X2 ,
θ
∇Y (f X) = Y f · X + f ∇Y X,
for θ = 1, 2 and X, Y, X1 , X2 , Y1 , Y2 ∈ X(MN ), f ∈ F(MN ), where F (MN ) is an
1
2
algebra of smooth real functions on MN . In that case we write LN = (MN , ∇, ∇)
1
2
and LN call a space on non-symmetric connections ∇, ∇.
If we introduce local coordinates x1 , . . . , xN and put ∂/∂xi = ∂i , in view of
(1.1) it will be
(1.2)
2
1
∇∂j ∂k = ∇∂k ∂j .
2010 Mathematics Subject Classification: Primary 53C05; Secondary 53B05.
Key words and phrases: non-symmetric connection, curvature tensors, independent curvature
tensors, Ricci type identities.
205
206
MINČIĆ
1
Denoting coefficients of the connection ∇ in the base ∂1 , . . . , ∂N with Lijk , we have
1
2
1
∇∂k ∂j = Lijk ∂i , ∇∂k ∂j = ∇∂j ∂k = Likj ∂i , where = denotes “equal with respect
(1.2)
(1.2)
to (1.2)".
Further, if we take by definition
θ
θ
θ
T (X, Y ) = ∇Y X − ∇X Y + [X, Y ], θ ∈ {1, 2},
it follows
2
1
T (X, Y ) = −T (X, Y ) ≡ −T (X, Y ),
2
1
1
2
(T (X, Y ) = T (X, Y )) ⇔ (∇ = ∇ = ∇).
We proved in [18] how it is possible to obtain several curvature tensors in LN by
polylinear mappings. It is proved that among these tensors there are 5 independent
ones:
1
1
1
1
1
1
2
2
2
2
2
2
(1.3)
R(X; Y, Z) = ∇Z ∇Y X − ∇Y ∇Z X + ∇[Y,Z] X,
(1.4)
R(X; Y, Z) = ∇Z ∇Y X − ∇Y ∇Z X + ∇[Y,Z] X,
3
2
1
1
2
2
(1.5)
R(X; Y, Z) = ∇Z ∇Y X − ∇Y ∇Z X + ∇ 1
(1.6)
R(X; Y, Z) = ∇Z ∇Y X − ∇Y ∇Z X + ∇ 2
∇Y Z
4
2
1
1
2
2
∇Y Z
1
X − ∇2
∇Z Y
1
X − ∇1
∇Z Y
X,
X,
(1.7)
2
1
2
2
1
2
1
2
5
1 1 1
R(X; Y, Z) = (∇Z ∇Y X − ∇Y ∇Z X + ∇Z ∇Y X − ∇Y ∇Z X + ∇[Y,Z] X + ∇[Y,Z] X),
2
while the rest can be expressed as linear combinations of these five tensors. For
X = ∂/∂xj ≡ ∂j , Y = ∂k , Z = ∂l , one obtains
1
(1.8)
Rijkl = Lijk,l − Lijl,k +Lpjk Lipl − Lpjl Lipk ,
(1.9)
Rijkl = Likj,l − Lilj,k +Lpkj Lilp − Lplj Likp ,
(1.10)
Rijkl = Lijk,l − Lilj,k +Lpjk Lilp − Lplj Lipk +Lplk (Lipj − Lijp ),
(1.11)
Rijkl = Lijk,l − Lilj,k +Lpjk Lilp − Lplj Lipk +Lpkl (Lipj − Lijp ),
(1.12)
Rijkl =
2
3
4
5
1 i
(L +Likj,l −Lijl,k −Lilj,k +Lpjk Lipl +Lpkj Lilk −Lpjl Likp −Lplj Lipk )
2 jk,l
2. Identities for a vector and for a covector by both connections
2.1.
(2.1)
Consider an expression
ν
µ
µ
ν
(∇Z ∇Y X − ∇Y ∇Z X)(ω), µ, ν ∈ {1, 2}.
ON RICCI TYPE IDENTITIES...
207
Let us quote the relations (for µ = 1, 2)
µ
µ
µ
(∇Y X)(ω) = Y [X(ω)] − X(∇Y ω),
µ
(∇Y ω)(X) = Y [ω(X)] − ω(∇Y X).
and denote
µ
ν
b) ∇Z ω = ω ∈ X∗ (MN ).
a) ∇Y X = X ∈ X(MN ),
(2.2)
Then we have
µ
ν
ν
ν
(∇Z ∇Y X)(ω) = (∇Z X)(ω) = Z[X(ω)] − X(∇Z ω)
(2a)
(2)
µ
µ
= Z[(∇Y X)(ω)] − (∇Y X)(ω)
(2)
µ
(2.3)
µ
ν
ν
= Z{Y [X(ω)] − X(∇Y X)} − {Y [X(∇Z ω)] − X(∇Y ∇Z ω)}
(2b)
µ
µ
ν
ν
= ZY [X(ω)] − Z[X(∇Y ω)] − Y [X(∇Z ω)] + X(∇Y ∇Z ω),
and one gets
ν
(2.4)
µ
µ
ν
µ
ν
µ
ν
(∇Z ∇Y X − ∇Y ∇Z X)(ω) = [Z, Y ][X(ω)] − X(∇Z ∇Y ω − ∇Y ∇Z ω),
i.e.,
µ
ν
µ
ν
µ
ν
µ
ν
(∇Z ∇Y X − ∇Y ∇Z X)(ω) = [Z, Y ][X(ω)]−(∇Z ∇Y ω − ∇Y ∇Z ω)(X), µ, ν ∈ {1, 2}.
From
ν
ν
ν
ν
(2.5) (∇[Z,Y ] X)(ω) = ∇[Z,Y ] [X(ω)] − X(∇[Z,Y ] ω) = [Z, Y ][X(ω)] + (∇[Y,Z] ω)(X),
we find the first addend on the right side and substitute in (2.7). We obtain
ν
µ
ν
µ
ν
(2.6) (∇Z ∇Y X − ∇Y ∇Z X + ∇[Y,Z] X)(ω)
ν
µ
µ
ν
ν
= −(∇Z ∇Y ω − ∇Y ∇Z ω + ∇[Y,Z] ω)(X),
µ, ν ∈ {1, 2}
Definition 2.1. The equations (2.4) for µ, ν ∈ {1, 2} are Ricci type identities
for a vector in LN .
1
2.2.
(2.7)
Taking µ = ν = 1, we obtain the corresponding identity for ∇:
1
1
1
1
1
1
R(X; Y, Z)(ω) = −(∇Z ∇Y ω − ∇Y ∇Z ω + ∇[Y,Z] ω)(X).
Denoting
1
(2.8)
1
1
1
1
1
R(ω; Y, Z) = ∇Z ∇Y ω − ∇Y ∇Z ω + ∇[Y,Z] ω,
the equation (2.7) gives a relation
(2.9)
1
1
R(X; Y, Z)(ω) = −R(ω; Y, Z)(X).
208
MINČIĆ
In order to write the equation (2.4) in local coordinates for µ = ν = 1, we take
X = X j ∂j , Y = ∂k , Z = ∂l , ω = dxi . For the left side in (2.4) we obtain
1
1
1
1
L = (∇∂l ∇∂k X − ∇∂k ∇∂l X)(dxi )
1
1
= [∇∂l (X |jk ∂j ) − ∇∂k (X |jl ∂j )](dxi )
1
=
1
[(X |jk ),l ∂j
+
X |jk Lpjl ∂p
1
=
− (X |jl ),k ∂j − X |jl Lpjk ∂p ](dxi )
1
(X |jk ),l δji
+
1
X |jk Lpjl δpi
−
−
X |jl Lpjk δpi
1
1
1
1
(X |jl ),k δji
1
= (X |ik ),l + X |jk Lijl − (X |il ),k − X |jl Lijk
1
=
+
Lpkl X |ip
1 1
=
1
1
X |ik| l
−
X |il| k
1
X |ik| l
−
X |ik| l
1 1
+
1
−
Lplk X |ip
1 1
1
p
Tkl
X |ip .
1 1
1
For the right-hand side in (2.4) we obtain
1
1
1
1
R = [∂l , ∂k ](X(dxi )) + X(∇∂l ∇∂k dxi − ∇∂k ∇∂l dxi )
1
1
= 0 + X[∇∂l (−Lipl dxp ) − ∇∂l (−Lipk dxp )]
= X(Ripkl dxp ) = Ripkl dxp (X) = Ripkl X p ,
1
1
1
and from L = R, we have
p
X |ikl − X |ilk = Ripkl X p − Tkl
X |ip ,
(2.10)
1
1
1
1
i.e., the known identity in local coordinates. So, we have proved the following
theorem.
1
Theorem 2.1. In the space LN , with non-symmetric affine connection ∇ by
equation (2.4) for µ = ν = 1 the first Ricci type identity for a vector is given. That
1
1
identity can be written in forms (2.5), (2.6), (2.9), here R is given by (1.3) and R
by (2.8). The corresponding identity in local coordinates is (2.10).
2.3. By using equation (2.4) and the condition X(ω) = ω(X) ∈ F(MN ), we
obtain the equation analogous to (2.4) (X and ω have changed the roles):
(2.11)
ν
µ
µ
ν
ν
µ
µ
ν
(∇Z ∇Y ω − ∇Y ∇Z ω)(X) = [Z, Y ][ω(X)] − ω(∇Z ∇Y X − ∇Y ∇Z X).
Definition 2.2. Equations (2.11) for µ, ν ∈ {1, 2} are Ricci type identities for
a covector in LN .
Equation (2.11) can be obtained also by consideration of the expression on the
left-hand side in (2.11). The known Ricci identity for a covariant vector in local
ON RICCI TYPE IDENTITIES...
209
coordinates can be obtained from (2.11) by substituting ω = ωj xj , X = ∂i , Y = ∂k ,
Z = ∂l :
p
ωj | kl − ωj | lk = −Rpjkl ωp − Tkl
ωj | p .
(2.12)
1
1
1
1
So, the following theorem is valid.
1
Theorem 2.2. In the space LN , with non-symmetric affine connection ∇, by
equation (2.11) for µ = ν = 1, the first Ricci type identity for a covector is given.
The corresponding identity in local coordinates is (2.12).
For µ = ν = 2 from (2.4) is obtained
2.4.
2
(2.13)
2
2
2
2
2
2
2
(∇Z ∇Y X − ∇Y ∇Z X)(ω) = [Z, Y ][X(ω)] − X(∇Z ∇Y ω − ∇Y ∇Z ω).
From here
2
2
(2.14)
R(X; Y, Z)(ω) = −R(ω; Y, Z)(X),
2
2
2
where R is expressed by ∇ analogously to (2.8) and R is given in (1.3). Surpassing
to local coordinates, from (2.13) one obtains
p
X |ikl − X |ilk = Ripkl X p + Tkl
X |ip ,
(2.15)
2
2
2
2
and also equations similar to (2.11), (2.12) (for a covector).
Thus, we state
2
Theorem 2.3. In the space LN with non-symmetric affine connection ∇, defined by (1.1), the second Ricci type identity for a vector is given by equation (2.13).
The corresponding identity in local coordinates is (2.15).
3. Identities for a vector and covector
obtained by combinations of both connections
3.1.
Putting µ = 1, ν = 2 into (2.4), we get the identity
2
(3.1)
1
1
2
2
1
1
2
(∇Z ∇Y X − ∇Y ∇Z X)(ω) = [Z, Y ][X(ω)] − (∇Z ∇Y ω − ∇Y ∇Z ω)(X).
and from (2.6)
2
1
1
2
2
2
1
1
2
2
(3.2) (∇Z ∇Y X − ∇Y ∇Z X + ∇[Y,Z] X)(ω) = −(∇Z ∇Y ω − ∇Y ∇Z ω + ∇[Y,Z] ω)(X)
Analogously to (1.5), let us put
3
(3.3)
2
1
1
2
2
R(ω; Y, Z) = ∇Z ∇Y ω − ∇Y ∇Z ω + ∇ 1
∇Y Z
1
ω − ∇2
∇Y Z
ω ∈ X∗ (MN )
210
MINČIĆ
and (3.2) becomes
3
1
2
(3.4) (R(X; Y, Z) + ∇ 2
∇Y Z
2
X − ∇1
∇Y Z
X + ∇[Y,Z] X)(ω)
3
1
2
= −(R(ω; Y, Z) + ∇ 2
∇Z Y
ω − ∇1
2
∇Y Z
ω + ∇[Y,Z] ω)(X).
Because of
1
2
(∇ 2
∇Z Y
2
X − ∇1
∇Y Z
1
2
X + ∇[Y,Z] X)(ω) = (∇ 2
∇Z Y
1
X − ∇1
∇Y Z+[Z,Y ]
2
= (∇ 2
∇Z Y
X − ∇2
∇Z Y
X)(ω)
X)(ω)
and
1
−(∇ 2
2
∇Z Y
2
ω − ∇1
∇Y Z
1
ω + ∇[Y,Z] ω)(X) = −(∇ 2
∇Z Y
2
1
= (∇ 2
∇Z Y
2
ω − ∇2
∇Z Y
∇Z Y
1
[ω(X)] − ω(∇ 2
∇Z Y
2
= (∇ 2
∇Z Y
∇Y Z+[Z,Y ]
ω)(X)
ω)(X)
2
= ∇2
2
ω − ∇1
X − ∇2
∇Z Y
1
X) − ∇ 2
∇Z Y
1
[ω(X)] + ω(∇ 2
∇Z Y
X)
X)(ω),
we see that the right-hand sides of these equations are identical and from (3.4):
3
3
(3.5)
R(X; Y, Z)(ω) = −R(ω; Y, Z)(X).
3.2. If we put X = X j ∂j , Y = ∂k , Z = ∂l , ω = dxi , equation (3.1) will be
written in local coordinates as follows. For the left-hand side L we have
2
1
1
2
L = (∇∂l ∇∂k X − ∇∂k ∇∂l X)(dxi )
2
1
= [∇∂l (X |jk ∂j ) − ∇∂k (X |jl ∂j )](dxi )
1
(3.6)
2
= [(X |jk ),l ∂j + X |jk Lplj ∂p − (X |jl ),k ∂j − X |jl Lpjk ∂p ](dxi )
1
=
1
(X |ik ),l
+
1
=
X |ik| l
1 2
2
X |jk Lilj
−
(X |il ),k
−
−
2
1
X |il| k
2
X |jl Lijk
−
Lplk (X |ip
2 1
2
−
X |ip )
1
i
= X |ik| l − X |il| k − Lplk Tsp
X s,
2
1 2
2 1
where
X |ik| l = (X |ik ),l + X |pk Lilp − X |ip Lplk ,
1 2
X |il| k
2 1
1
=
(X |il ),k
2
1
+
X |pl Lipk
2
1
− X |ip Lplk .
2
ON RICCI TYPE IDENTITIES...
211
For the right-hand side one obtains
2
1
1
2
R = −(∇∂l ∇∂k dxi − ∇∂k ∇∂l dxi )(X)
(3.7)
= (Lipk,l dxp − Lipk Lpls dxs + Lilp,k dxp + Lilp Lpsk dxs )(X)
= (Lipk,l − Lisk Lslp − Lilp,k + Lils Lspk )X p .
By virtue of (3.6) and (3.7), from L = R it is
X |ik| l − X |il| k = Ripkl X p ,
(3.8)
1 2
3
2 1
and, analogously to that exposed above, for a covariant vector ω it is obtained
ωj | k| l − ωj | l| k = −Rpjkl ωp .
(3.9)
1 2
3
2 1
4
4
3.3. Introducing R(X; Y, Z) into (3.2) by virtue of (1.6) and defining R(ω; Y, Z)
according to
4
(3.10)
2
1
1
2
2
1
R(ω; Y, Z) = ∇Z ∇Y ω − ∇Y ∇Z ω + ∇ 2
∇Y Z
ω − ∇1
∇Y Z
ω ∈ X∗ (MN ),
equation (3.2) gives
4
1
2
(R(X; Y, Z) + ∇ 1
∇Z Y
4
X − ∇2
1
∇Y Z
2
= −(R(ω; Y, Z) + ∇ 1
∇Z Y
ω − ∇2
∇Y Z
2
X + ∇[Y,Z] X)(ω)
2
ω + ∇[Y,Z] ω)(X).
3
As in the case of R, we obtain
4
4
(3.11)
R(X; Y, Z)(ω) = −R(ω; Y, Z)(X).
Putting X = ∂j , Y = ∂k , Z = ∂l , ω = dxi and taking into consideration (3.10), we
get
4
2
2
2
1
Rpjkl ∂p (dxi ) = −[∇∂l ( − Lipk dxp ) − ∇∂k ( − Lilp dxp ) + ∇Lpkl ∂p dxi − ∇Lpkl ∂p dxi ](∂j ),
4
from where for R the value (1.11) is obtained. In view of (1.10), (1.11) it is
4
3
p
i
Rpjkl − Rpjkl = Tpj
Tkl
.
and, using (3.8) and (3.9), we obtain
(3.12)
i
s
Tkl
X p,
X |ik| l − X |il| k = Ripkl X p + Tps
1 2
(3.13)
2 1
4
p s
ωj | k| l − ωj | l| k = −Rpjkl ωp + Tsj
Tkl ωp .
1 2
2 1
Now, we can state the following theorem
4
212
MINČIĆ
1
Theorem 3.1. In the space LN with two non-symmetric affine connections ∇,
2
∇, linked by equation (1.1), the third Ricci type identity for a vector is given by
equation (3.1). This identity can be written also in forms (3.2), (3.5) and (3.11).
From (2.8), for µ = 1, ν = 2, one obtains the third Ricci type identity for a
covector. The corresponding identities in coordinates are (3.8), (3.9), (3.12) and
(3.13).
5
3.4. In order to obtain an identity in which R appears, let us start from the
expression which appears in (1.7). So,
1
1
2
2
1
2
2
1
(∇Z ∇Y X + ∇Z ∇Y X − ∇Y ∇Z X − ∇Y ∇Z X)(ω)
1
1
1
1
(2.3) = ZY [X(ω)] − Z[X(∇Y ω)] − Y [X(∇Z ω)] + X(∇Y ∇Z ω)
2
2
2
2
2
1
2
1
+ ZY [X(ω)] − Z[X(∇Y ω)] − Y [X(∇Z ω)] + X(∇Y ∇Z ω)
− Y Z[X(ω)] + Y [X(∇Z ω)] + Z[X(∇Y ω)] − X(∇Z ∇Y ω)
1
2
1
2
− Y Z[X(ω)] + Y [X(∇Z ω)] + Z[X(∇Y ω)] − X(∇Z ∇Y ω) (ω),
that is
(3.14)
2
2
1
2
2
1
1 1 1
(∇Z ∇Y X + ∇Z ∇Y X − ∇Y ∇Z X − ∇Y ∇Z X)(ω)
2
1
1
2
2
1
2
2
1
1
= [Z, Y ][X(ω)] + X(∇Y ∇Z ω + ∇Y ∇Z ω − ∇Z ∇Y ω − ∇Z ∇Y ω).
2
Definition 3.1. Equation (3.14) we call the combined Ricci type identity for
a vector in LN .
Using (1.7), from (3.14) it is obtained
5
0
(3.15) R(X; Y, Z)(ω)+(∇[Z,Y ] X)(ω)
2
2
1
2
2
1
1 1 1
= [Z, Y ][ω(X)]+ (∇Y ∇Z ω + ∇Y ∇Z ω − ∇Z ∇Y ω − ∇Z ∇Y ω)(X).
2
From
0
0
0
(∇[Z,Y ] X)(ω) = [Z, Y ][X(ω)] − X(∇[Z,Y ] ω) = [Z, Y ][ω(X)] + (∇[Y,Z] ω)(X),
we find the first addend of the right-hand side and substitute into (3.15). So,
5
0
R(X; Y, Z)(ω) + (∇[Z,Y ] X)(ω)
0
0
= (∇[Z,Y ] X)(ω) + (∇[Z,Y ] ω)(X)
2
2
1
2
2
1
1 1 1
+ (∇Y ∇Z ω + ∇Y ∇Z ω − ∇Z ∇Y ω − ∇Z ∇Y ω)(X),
2
ON RICCI TYPE IDENTITIES...
213
5
where R is given in (1.7). Denoting
(3.16)
5
2
2
1
2
2
1
1
2
1 1 1
R(ω; Y, Z) = (∇Z ∇Y ω + ∇Z ∇Y ω − ∇Y ∇Z ω − ∇Y ∇Z ω + ∇[Y,Z] ω + ∇[Y,Z] ω)
2
the previous equation gives
5
5
(3.17)
R(X; Y, Z)(ω) = R(ω; Z, Y )(X).
Substituting here X = ∂j , Y = ∂k , Z = ∂l , ω = dxi and taking into consideration
5
(3.16), for Rijkl (1.12) is obtained.
5
5
Remark 3.1. We see that relation between R and R is not of the form relating
θ
θ
to R, R, θ = 1, 2, 3, 4. In fact using the corresponding values from [22]
5
0
8
2
1
1
1
2
2
1
2
1 1 2
(∇Z ∇Y X + ∇Z ∇Y X − ∇Y ∇Z X − ∇Y ∇Z X + ∇[Y,Z] X + ∇[Y,Z] X)
2
R = R(X; Y, Z) + τ (τ (X, Y ), Z) + τ (τ (X, Z), Y ),
R=
0
= R(X; Y, Z) − τ (τ (X, Y ), Z) − τ (τ (X, Z), Y ).
5
0
8
We conclude that R − 2R = R and
5
5
8
R(X; Y, Z)(ω) = R(ω; Z, Y )(X) = −R(ω; Y, Z)(X).
So, we have
1
2
Theorem 3.2. In the space LN with two non-symmetric affine connections ∇,
∇, linked according to (1.1), by equation (3.14) the combined Ricci type identity
for a vector is given. Some other forms of (3.14) are (3.15)–(3.17). From (3.15)
combined Ricci type identity for a covector is obtained:
2
2
1
2
2
1
1 1 1
(∇Y ∇Z ω + ∇Y ∇Z ω − ∇Z ∇Y ω − ∇Z ∇Y ω)(X)
2
2
2
1
2
2
1
1 1 1
= (∇Z ∇Y X + ∇Z ∇Y X − ∇Y ∇Z X − ∇Y ∇Z X)(ω) + [Z, Y ][ω(X)].
2
From (3.14) and (3.18) we obtain the corresponding combined Ricci type identities for a vector and covector respectively local coordinates:
5
1 i
(X + X |ikl − X |il| k − X |il| k ) = Ripkl X p ,
2 1| kl
2
1 2
2 1
(3.18)
5
0
1
(ωj | kl + ωj | kl − ωj | l| k − ωj | l| k ) = (R − 2R)pjkl ωp ,
2 1
2
1 2
2 1
0
0
where Rijkl is defined by Lijk = 21 (Lijk + Lijk ), i.e., by symmetric connection coefficients.
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MINČIĆ
θ
θ
Definition 3.2. The objects R, (θ = 1, . . . , 5), defined by R : X × X × X → X∗
θ
on MN , we call dual curvature tensors in relation to R.
4. Identities for a tensor field t of the type (r, s)
Let us consider a tensor field of the type (r, s), which will be denoted
4.1.
r
r
t ≡ t, i.e., consider a mapping t : (X ∗ )r × (X )s 7→ F (MN ). So,
s
s
r
t(ω 1 , . . . , ω r , X1 , . . . , Xs ) ∈ F(MN ),
s
is a differentiable function on MN .
As known, a covariant derivative
µ
ν
r
∇Y
µ
r
t is also of a type (r, s). As in (2.1), one
s
ν
r
can consider the expression (∇Z ∇Y t − ∇Y ∇Z t)(ω 1 , . . . , ω r , X1 , . . . , Xs ).
s
s
2
Let us examine nearer the case (r, s) = (2, 1), i.e., t ≡ t. We have
4.2.
1
2
µ
µ
µ
(∇Y t)(ω 1 , ω 2 ; X) = ∇Y [t(ω 1 , ω 2 ; X] − t(∇Y ω 1 , ω 2 ; X)
1
µ
µ
− t(ω 1 , ∇Y ω 2 ; X) − t(ω 1 , ω 2 ; ∇Y X).
µ
Denoting ∇Y t = t, we have
µ
ν
ν
(∇Z ∇Y t)(ω 1 , ω 2 ; X) = (∇Z t)(ω 1 , ω 2 ; X)
ν
ν
ν
ν
= ∇Z [t(ω 1 , ω 2 ; X] − t(∇Z ω 1 , ω 2 ; X) − t(ω 1 , ∇Z ω 2 ; X) − t(ω 1 , ω 2 ; ∇Z X)
(4)
ν
µ
µ
µ
ν
µ
ν
ν
= ∇Z [(∇Y t)(ω 1 , ω 2 ; X] − (∇Y t)(∇Z ω 1 , ω 2 ; X) − (∇Y t)(ω 1 , ∇Z ω 2 ; X) − (∇Y t)(ω 1 , ω 2 ; ∇Z X)
ν
µ
µ
µ
µ
= ∇Z {∇Y [t(ω 1 , ω 2 ; X] − t(∇Y ω 1 , ω 2 ; X) − t(ω 1 , ∇Y ω 2 ; X) − t(ω 1 , ω 2 ; ∇Y X)}
(4)
µ
µ
ν
ν
µ
ν
µ
ν
− {∇Y [t(∇Z ω 1 , ω 2 ; X] − t(∇Y ∇Z ω 1 , ω 2 ; X) − t(∇Z ω 1 , ∇Y ω 2 ; X) − t(∇Z ω 1 , ω 2 ; ∇Y X)}
µ
µ
ν
µ
ν
ν
µ
ν
− {∇Y [t(ω 1 , ∇Z ω 2 ; X] − t(∇Y ω 1 , ∇Z ω 2 ; X) − t(ω 1 , ∇Y ∇Z ω 2 ; X) − t(ω 1 , ∇Z ω 2 ; ∇Y X)}
µ
µ
ν
µ
ν
µ
ν
ν
− {∇Y [t(ω 1 , ω 2 ; ∇Z X] − t(∇Y ω 1 , ω 2 ; ∇Z X) − t(ω 1 , ∇Y ω 2 ; ∇Z X) − t(ω 1 , ω 2 ; ∇Y ∇Z X)}.
wherefrom
(4.1)
ν
µ
2
µ
ν
2
2
(∇Z ∇Y t − ∇Y ∇Z t)(ω 1 , ω 2 ; X) = [Z, Y ][t(ω 1 , ω 2 ; X)]
1
1
1
2 ν
µ
1
2
ν
µ
ν
= −t(∇Z ∇Y ω 1 − ∇Y ∇Z ω 1 , ω 2 ; X)
µ
µ
ν
− t(ω 1 , ∇Z ∇Y ω 2 − ∇Y ∇Z ω 2 ; X)
1
2
ν
µ
µ
ν
− t(ω 1 , ω 2 ; ∇Z ∇Y X − ∇Y ∇Z X).
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ON RICCI TYPE IDENTITIES...
215
In the general case, starting from
4.3.
µ
µ
r
r
(∇Y t)(ω 1 , . . . , ω r ; X1 , . . . , Xs ) = ∇Y [t(ω 1 , . . . , ω r ; X1 , . . . , Xs )]
(4.2)
s
s
r
X
µ
r
−
t(ω 1 , . . . , ω i−1 , ∇Y ω i , ω i+1 , . . . , ω r ; X1 , . . . , Xs )
−
i=1
s
X
j=1
s
µ
r
t(ω 1 , . . . , ω r ; X1 , . . . , Xj−1 , ∇Y Xj , Xj+1 , . . . , Xs ),
s
we get
µ
ν
µ
r
ν
r
s
s
r
(∇Z ∇Y t − ∇Y ∇Z t)(ω 1 , . . . , ω r ; X1 , . . . , Xs ) = [Z, Y ][t(ω 1 , . . . , ω r ; X1 , . . . , Xs )]
s
(4.3)
−
r
X
µ
µ
ν
ν
r
t(ω 1 , . . . , ω i−1 , ∇Z ∇Y ω i − ∇Y ∇Z ω i , ω i+1 , . . . , ω r ; X1 , . . . , Xs )
−
i=1
s
X
j=1
s
ν
r
µ
µ
ν
t(ω 1 , . . . , ω r ; X1 , . . . , Xj−1 , ∇Z ∇Y Xj − ∇Y ∇Z Xj , Xj+1 , . . . , Xs ).
s
Consider some particular cases, obtained from (4.2). For example:
1
1) For µ = ν = 1, r = 1, s = 0 that is for t = X, from (2.4) corresponding
0
identity is obtained, i.e., in coordinates, (2.10), and for r = 1, s = 0 it
follows (2.11), respectively (2.12)
2) For µ = ν = 2, r = 1, s = 0 analogous relation (2.13), and equations
corresponding to (2.11) and (2.12) are obtained.
3) For µ = 1, ν = 2, r = 1, s = 0 we obtain (3.1) and for r = 0, s = 1 the
corresponding equation follows, where the roles of X and ω are exchanged.
4) For r = 2, s = 1, relation (4.1) follows.
4.4.
Identities (4.3) can be written so that in them curvature tensors figure
1
explicitly. For example, for µ = ν = r = s = 1, ∇ ≡ ∇ we have
1
1
1
1
1
(∇Z ∇Y t − ∇Y ∇Z t)(ω; X) = [Z, Y ][t(ω; X)]
1
1
1
− t(∇Z ∇Y ω − ∇Y ∇Z ω; X) − t(ω; ∇Z ∇Y X − ∇Y ∇Z X)
1
1
1
1 1
1
1
1
1 1
1
1
1
1
1
1
1
1
= ∇[Z,Y ] [t(ω; X)] − t(R(ω; Y, Z) − ∇[Y,Z] ω; X) − t(ω; R(X; Y, Z) − ∇[Y,Z] X)
1
1
1
= ∇[Z,Y ] [t(ω; X)] − t(R(ω; Y, Z); X) + t(∇[Y,Z] ω; X) − t(ω; R(X; Y, Z)) + t(ω; ∇[Y,Z] X).
1
In view of (4.2) it is
1
1
1
1
1
1
1
1
(∇[Z,Y ] t)(ω; X) = ∇[Z,Y ] [t(ω; X)] + t(∇[Y,Z] ω; X) + t(ω; ∇[Y,Z] X),
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MINČIĆ
the previous equation gives the identity
1
1
1
1
1
1
(∇Z ∇Y t − ∇Y ∇Z t)(ω; X)
1
(4.4)
1
1
1
1 1
1
1
1
1
1
= (∇[Z,Y ] t)(ω; X)] − t(R(ω; Y, Z); X) − t(ω; R(X; Y, Z)).
Herefrom, in the local coordinates one obtains
1
1
p i
tij | kl − tij | lk = Ripkl tpj − Rpjkl tip − Tkl
tj | p .
1
1
1
Finally, from the exposed, the following theorem is valid.
1
2
Theorem 4.1. In the space LN with two non-symmetric connections ∇, ∇,
linked with equation (1.1), equation (4.3) represents general Ricci type identity for
r
a tensor t of the type (r, s). The equations obtained previously for a vector and a
s
covector, also (4.4), are particular cases of (4.3).
θ
θ
In (4.3) we see how the quantities R and R can be introduced.
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Faculty of Science and Mathematics
University of Niš
Niš
Serbia
svetislavmincic@yahoo.com